Consider the differential equation
⦁ A solution curve passes through the point . What is its slope at this point?
⦁ Argue that every solution curve is increasing for .
⦁ Show that the second derivative of every solution satisfies
⦁ A solution curve passes through (0,0). Prove that this curve has a relative minimum at (0,0).
⦁ The slope at the point 2.
⦁ Yes, every solution curve is increasing for x > 1.
⦁ The second derivative of every solution satisfies the given equation.
⦁ Yes, the curve has a minimum at (0,0)
Slope is given by
So, slope of the solution curve at is
Hence, the slope at the point is 2.
Since, for all
Then, , for all .
i.e., , for all , .
Hence, from first derivative test, every solution curve is increasing for .
Differentiate both sides with respect to x
So, the second derivative of every solution satisfies the given equation.
we get, at
thus, is a critical point.
Also, from part (c) ,
From second derivative test, is a point of relative minimum.
Therefore, the curve has a minimum at
In Problems 14–24, you will need a computer and a programmed version of the vectorized classical fourth-order Runge–Kutta algorithm. (At the instructor’s discretion, other algorithms may be used.)†
Using the vectorized Runge–Kutta algorithm with h = 0.5, approximate the solution to the initial value problemat t = 8.
Compare this approximation to the actual solution .
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